﻿using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Numerics;

namespace ProjectEulerSolutions
{
    /*
     * The square root of 2 can be written as an infinite continued fraction.
√2 = 1 + 	
1

  	2 + 	
1

  	  	2 + 	
1

  	  	  	2 + 	
1

  	  	  	  	2 + ...

The infinite continued fraction can be written, √2 = [1;(2)], (2) indicates that 2 repeats ad infinitum. In a similar way, √23 = [4;(1,3,1,8)].

It turns out that the sequence of partial values of continued fractions for square roots provide the best rational approximations. Let us consider the convergents for √2.
1 + 	
1

	= 3/2
  	
2
	 
1 + 	
1

	= 7/5
  	2 + 	
1

  	  	
2
	 
1 + 	
1

	= 17/12
  	2 + 	
1

	 
  	  	2 + 	
1

	 
  	  	  	
2
	 
1 + 	
1

	= 41/29
  	2 + 	
1

  	  	2 + 	
1

	 
  	  	  	2 + 	
1

	 
  	  	  	  	
2
	 

Hence the sequence of the first ten convergents for √2 are:
1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, ...

What is most surprising is that the important mathematical constant,
e = [2; 1,2,1, 1,4,1, 1,6,1 , ... , 1,2k,1, ...].

The first ten terms in the sequence of convergents for e are:
2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, ...

The sum of digits in the numerator of the 10th convergent is 1+4+5+7=17.

Find the sum of digits in the numerator of the 100th convergent of the continued fraction for e.

     * */
    class Problem65 : IProblem
    {
        public string Calculate()
        {
            //3 milisekunde... 

            //uzmemo n-tu znamenku u CF-u, ona je prvi denom
            //1 je prvi nom
            //uzmemo n-1-tu znamenku, ona je prvi whole

            //denom = whole + 1 / denom

            //dakle nom = denom;
            //dakle denom = (whole * denom + 1)
            
            int index = 100;

            //oooooogromni brojevi...
            BigInteger nom = 0;
            BigInteger denom = 1;

            while (index >= 1)
            {
                //ovim je definiran continued fraction za e [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, ... , 1, 2k, 1...]
                //modulo 3 pozicije su sa vrijednosti 2*index/3;, prva pozicija je 2, ostale su 1
                int whole = 0;
                if (index == 1)
                    whole = 2;
                else if (index % 3 == 0)
                    whole = 2 * index / 3;
                else
                    whole = 1;

                BigInteger tempNom = nom;
                nom = denom;
                denom = whole * denom + tempNom;
                index--;
            }

            Console.WriteLine("{0}/{1}", denom, nom);

            return CommonFunctions.GetDigits(denom).Sum().ToString();
        }
    }
}
